shown in the last column of Figure 8.14 are obtained, which are degenerate in
the axial index n for the infinite cylinder.
As b/a increases from 1.111… to 4.0, the prolate spheroid levels (l,m,s) of
different l values
,...)1,( mml but with the same value of m and s are seen
to collect together. These families of different l values are illustrated in Figure
8.14 by lines of various dash lengths. As these levels collect together, they are
also seen to be nesting down a cylindrical family (n,m,s). The nesting of the
latter is indicated by solid lines. This can be justified by noting that as
foab / , the radial prolate spheroidal functions tend towards the Bessel
functions of the cylinder. Similarly as
1/ oab , the radial functions tend
towards the spherical Bessel functions.
Figure 8.14 showed that the shifts and splitting of the eigenfrequencies
under deformations of the target provide considerable information for shape
determination of acoustic targets.
8.5.3 Material Characterization by Resonance Acoustic
Spectroscopy (MCRAS)
In this section, an approach in resonance acoustic spectroscopy proposed by
Honarvar and Sinclair [75] is presented. This method can be used for
verifying the elastic constants of an isotropic elastic rod. The accuracy of this
technique is verified by comparing it with other techniques. Experimental
results on the identification of the mode number n of different resonances of
an aluminum cylinder are also presented in this section.
It has been shown by previous researchers [ 40, 27, 80] that there exists a
good agreement between numerically calculated and experimentally measured
form functions for isotropic elastic cylinders. The elastic properties of the
cylinder required for calculating its numerical form function are the density
and two elastic constants of the material. These elastic properties can be
obtained by matching a numerical form function to the experimentally
measured form function. Although solving the inverse scattering problem
(ISP) seems to be the desired approach to finding the elastic constants of the
cylinder, application of ISP in practice is difficult. In this section, an
for extraction of the elastic properties of an isotropic elastic cylinder from its
measured form function.
In MCRAS, the elastic properties of a solid elastic cylinder are found by
matching the corresponding resonance frequencies on the measured and
calculated form functions through an iterative computer algorithm.
The logic behind the iterative program used for matching the resonances of
the measured form function to those of a numerically calculated form function
is briefly described here. Figure 8.15 shows the form function of an aluminum
rod calculated for a normally incident wave, i.e.,
Į 0
q
. Each resonance
8.5 Experimental and Numerical Results
381
alternative approach called material characterization by RAS (MCRAS) is used